Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23860 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | C22×D4 | D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×D4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
696 conjugacy classes of subgroups; only the 147 normal subgroups are shown.
This group has 696 conjugacy classes of subgroups — too many to draw. Only its 147 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(2,3)(5,6)(7,8)(9,10,15,11)(12,20,21,24)(13,18,14,17)(16,23,19,22);(1,2,4,3)(9,11,15,10)(12,20,21,24)(13,17,14,18)(16,23,19,22);(6,8)(9,12,15,21)(10,16,11,19)(13,23,14,22)(17,24,18,20);(1,3,4,2)(12,21)(16,19)(20,24)(22,23)Polycyclic presentation:
2^8|a^2=h,b^2=eh,c^2=h,d^2=e,f^2=h,g^2=h,b^a=be,c^a=cf,d^a=de,f^a=fh,g^a=gh,c^b=cg,f^b=fh,g^b=gh,d^c=dh,f^c=fh,g^c=gh46 conjugacy classes — view the full character table.
| maximal subgroup of | #250589, #250593, #250606, … |
| maximal normal subgroup of | #250589, #250593, #250606, … |
| maximal quotient of | #261354, #261365, #261374, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 17243);